Q-Learning [1]} maintains an estimate of the optimal value function \(Q: \mathcal {S} \times \mathcal {A} \rightarrow \mathbb {R}\)  which is updated towards the target:
\(\begin{split}Q(s_t, a_t) &\leftarrow Q(s_t, a_t) + \\&+ \alpha [r_{t+1} + \gamma {max}_{a^{^{\prime }}} Q(s_{t+1}, a^{^{\prime }})- Q(s_t, a_t)]\end{split}\) 
